$\begingroup$If you don't mind a wikipedia quote: " is used both in the SI world as well as where Imperial units are used. Its chief advantages are that its units and numerical value are identical everywhere"$\endgroup$
– Damien_The_UnbelieverJun 5 '15 at 6:52

Expressing specific impulse as a speed (Ve) is just as valid -- this effectively lets you use propellent mass alone vs. mass x g in the various equations.

Dividing by g0 is just a way of scaling the value and expressing specific impulse in seconds -- this effectively lets you use propellent weight in the various equations. This can be easier for some applications.

$\begingroup$I think I see. For example when calculating thrust (F) by multiplying the Isp by the mass flow rate. The Isp in seconds becomes handy then, and wouldn't work unless g0 was already factored in. I think I said that right.$\endgroup$
– David FourmanJun 4 '15 at 14:38

$\begingroup$Historically, it has been easier to talk about the weight of propellent versus the mass of propellent. So putting g into these equations is pretty much standard.$\endgroup$
– ErikJun 4 '15 at 14:44

Specific impulse is simply ratio of thrust to propellant consumption (mass flow) as have been mentioned. Traditionally, the force is measured by weight of unit mass (pound, kilogram). Related unit often are referred as pound force (kpf) or kilogram force (kgf). If the trust is measured by, kpf and consumption in lb/second dimension of specific impulse will be in seconds. Therefore, the answer would be g came from definition of force as weight of unit mass. In SI Isp is measured in m/sec i.e. theoretic exhaust velocity. The reason of sec to be preferred over m/sec or ft/sec is answered in previous posts.